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Your web server is sending a HTTP header, which tells the browser, which encoding the document is in. Most likely, it say either UTF-8 or ISO-8859-1. Both encodings are suitable for Danish characters (And western European in general).
If your web server says the document is encoded as UTF-8, but you do in fact send ISO-8859-1 encoded data, the browser will interpret the text wrongly. Since the two encodings are interchangeable on the US-ASCII range (codepoint 0-127), you will only see the problems with special characters (above the 127-range). These include the Danish national characters.

Most likely, your page is in ISO-8859-1, but you're advertising it as UTF-8. You can find out if this is the case, by opening the menu View > Charcter Encoding (In Firefox). The encoding, which the browser is using, will be highlighted.
To change it, you need to send the correct header, specifying which charset you're using. To specify ISO-8859-1, you would use the following:

Since you aren't sending any HTTP-header, specifying the charset, the meta-tag is used instead. This page is interpreted as UTF-8. That means that any strings, you output should be in UTF-8. Looking at the page, that seems to be the case.
Where do you get the malformed string from? Does it come from the database, or from a PHP string constant?